Bookmark

Mathematics > Group Theory

Title:
Palindromic words in simple groups

Abstract: A palindrome is a word that reads the same left-to-right as right-to-left. We
show that every simple group has a finite generating set $X$, such that every
element of it can be written as a palindrome in the letters of $X$. Moreover,
every simple group has palindromic width $pw(G,X)=1$, where $X$ only differs by
at most one Nielsen-transformation from any given generating set. On the
contrary, we prove that all non-abelian finite simple groups $G$ also have a
generating set $S$ with $pw(G,S)>1$.
As a by-product of our work we also obtain that every just-infinite group has
finite palindromic width with respect to a finite generating set. This provides
first examples of groups with finite palindromic width but infinite commutator
width.